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{{Infobox ET}}
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo|53 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 0.1166 cents stretched and the step size is about 22.6437 cents. It is consistent to the [[9-odd-limit|10-integer-limit]].
{{ED intro}}


Lookalikes: [[53edo]], [[84edt]]
== Theory ==
=Just Approximation=
31edf is almost identical to [[53edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1]] being [[just]]. The octave is [[stretched and compressed tuning|stretched]] by about 0.117 [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit|10-integer-limit]]. While the [[3-limit]] part is tuned sharp plus a sharper [[7/1|7]], the [[5/1|5]], [[11/1|11]], [[13/1|13]], and [[19/1|19]] remain flat but significantly less so than in 53edo.
31edf provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
{| class="wikitable"
|-
! |interval
! |ratio
! |size
! |difference
|-
| |perfect octave
| |2/1
| style="text-align:center;" |31
| | +0.12 cents
|-
| |major third
| |5/4
| style="text-align:center;" |17
| |−1.37 cents
|-
| |minor third
| |6/5
| style="text-align:center;" |14
| | +1.37 cents
|-
| |major tone
| |9/8
| style="text-align:center;" |9
| |−0.12 cents
|-
| |minor tone
| |10/9
| style="text-align:center;" |8
| |−1.25 cents
|-
| |diat. semitone
| |16/15
| style="text-align:center;" |5
| | +1.49 cents
|}One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.


The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.85 cents away from the just ratio 7/4, so 31EDF can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.
The [[The Riemann zeta function and tuning|local zeta peak]] around 53 is located at 52.996829, which has the octave stretched by 0.0718{{c}}; the octave of 31edf comes extremely close (differing by only {{sfrac|1|22}}{{c}}), thus minimizing relative error as much as possible.
[[Category:Edf]]
 
[[Category:Edonoi]]
=== Harmonics ===
{{Harmonics in equal|31|3|2|intervals=integer}}
{{Harmonics in equal|31|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31edf (continued)}}
 
=== Subsets and supersets ===
31edf is the 11th [[prime equal division|prime edf]], following [[29edf]] and coming before [[37edf]]. It does not contain any nontrivial subset edfs.
 
== See also ==
* [[9ed9/8]] – relative ed9/8
* [[53edo]] – relative edo
* [[84edt]] – relative edt
* [[137ed6]] – relative ed6
 
[[Category:53edo]]
[[Category:Zeta-optimized tunings]]

Latest revision as of 13:50, 18 June 2025

← 30edf 31edf 32edf →
Prime factorization 31 (prime)
Step size 22.6437 ¢ 
Octave 53\31edf (1200.12 ¢)
(convergent)
Twelfth 84\31edf (1902.07 ¢)
(convergent)
Consistency limit 10
Distinct consistency limit 10

31 equal divisions of the perfect fifth (abbreviated 31edf or 31ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 31 equal parts of about 22.6 ¢ each. Each step represents a frequency ratio of (3/2)1/31, or the 31st root of 3/2.

Theory

31edf is almost identical to 53edo, but with the perfect fifth rather than the 2/1 being just. The octave is stretched by about 0.117 cents. Like 53edo, 31edf is consistent to the 10-integer-limit. While the 3-limit part is tuned sharp plus a sharper 7, the 5, 11, 13, and 19 remain flat but significantly less so than in 53edo.

The local zeta peak around 53 is located at 52.996829, which has the octave stretched by 0.0718 ¢; the octave of 31edf comes extremely close (differing by only ⁠1/22⁠ ¢), thus minimizing relative error as much as possible.

Harmonics

Approximation of harmonics in 31edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.12 +0.12 +0.23 -1.14 +0.23 +5.09 +0.35 +0.23 -1.02 -7.52 +0.35
Relative (%) +0.5 +0.5 +1.0 -5.0 +1.0 +22.5 +1.5 +1.0 -4.5 -33.2 +1.5
Steps
(reduced) 		53
(22) 		84
(22) 		106
(13) 		123
(30) 		137
(13) 		149
(25) 		159
(4) 		168
(13) 		176
(21) 		183
(28) 		190
(4)
Approximation of harmonics in 31edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.36 +5.20 -1.02 +0.47 +8.73 +0.35 -2.68 -0.90 +5.20 -7.40 +6.22 +0.47
Relative (%) -10.4 +23.0 -4.5 +2.1 +38.6 +1.5 -11.8 -4.0 +23.0 -32.7 +27.5 +2.1
Steps
(reduced) 		196
(10) 		202
(16) 		207
(21) 		212
(26) 		217
(0) 		221
(4) 		225
(8) 		229
(12) 		233
(16) 		236
(19) 		240
(23) 		243
(26)

Subsets and supersets

31edf is the 11th prime edf, following 29edf and coming before 37edf. It does not contain any nontrivial subset edfs.

See also

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